Biology 427 BiomechanicsLecture 16. Basic fluid dynamics II: continuity and Bernoulli’s principle.•Recap definition of a fluid and Newton’s law of viscosity and a coevolutionary story.

•Conservation of mass and continuity•An application of continuity•Conservation of energy and Bernoulli’s principle•Applications of Bernoulli’s principle

2 rules govern flow•Newton’s Law of viscosity

•No slip condition (fluid adheres to solid surfaces)

τ = µ du/dy = ν d(ρ u)/dy

Blood viscosity depends non-linearly on red blood cell concentration ( hematocrit) and Temperature

5

4

3

2

10.1 0.2 0.3 0.4 0.5 0.6

0 10

20

3040

Relativebloodviscosity(relative to water)

O2 carrying capacity

Flow rate

2 more rules govern flow-- and applications of those rules to biology:•Conservation of mass -- the Principle of Continuity•Conservation of energy -- Bernoulli’s Principle

Volumein/timeuin Ain

Volumeout/timeuout Aout

Conservation of mass -- the “principle of continuity”: rate of mass flow in must equal rate of mass flow out.

Volumein/timeuin Ain

Conservation of mass -- the “principle of continuity”: rate of mass flow in must equal rate of mass flow out.

Σuin Ain = Σ uout Aout

Aout

Aout

Ain

Area 1 cm2 Blood velocity 10 cm/s

Aorta

Capillary Area 2 10-7 cm2

Blood velocity 0.1 cm/s

How many capillaries do we have?

Conservation of Energy: what is the relationship between fluid motion and pressure?

Conservation of Energy: what is the relationship between fluid motion and pressure?

Like mass, energy cannot appear or disappear unaccountably -- it is conserved. We will follow this along a streamline.

3 assumptions:flow is steady (no acceleration at a point in space, du/dt = 0fluid is incompressible (ρ = constant)flow is “inviscid” (µ= 0)

u1,h1,P1u2,h2,P2

Conservation of Energy: what is the relationship between fluid motion and pressure?

Along a stream line we consider three forms of energyPotential energy (PE = m g h)Kinetic energy (KE = m u2/2)Mechanical work (W = F d = P Area d = P V = P m/ ρ)

Along a streamline PE + KE + W = constant

u1,h1,P1u2,h2,P2

Conservation of Energy: what is the relationship between fluid motion and pressure?

Along a stream line we consider three forms of energyPotential energy (PE = g h)Kinetic energy (KE = u2/2)Mechanical work (W = F d = P Area d = P V = P / ρ)

Along a streamline (PE + KE + W)/m = constant

u1,h1,P1u2,h2,P2

Conservation of Energy: what is the relationship between fluid motion and pressure?

Along a streamline (PE + KE + W)/m = constant

u1,h1,P1u2,h2,P2

P2/ ρ + gh2 + u22/2 = P1/ ρ + gh1 + u1

2/2 (P2 - P1) /ρ + g(h2 -h1) + (u2

2- u12 )/2 = 0

(P2 - P1) /ρ + (u22- u1

2 )/2 = 0

Conservation of Energy: what is the relationship between fluid motion and pressure? (P2 - P1) /ρ + (u2

2- u12 )/2 = 0

Why is fluid faster over the top?

Conservation of Energy: what is the relationship between fluid motion and pressure? (P2 - P1) /ρ + (u2

2- u12 )/2 = 0

u1 = 0.2 m/s ; u2 = 3 m/s What is (P2 - P1) ?

Conservation of Energy: what is the relationship between fluid motion and pressure? (P2 - P1) /ρ + (u2

2- u12 )/2 = 0

Conservation of Energy: what is the relationship between fluid motion and pressure? (P2 - P1) /ρ + (u2

2- u12 )/2 = 0

Conservation of Energy: what is the relationship between fluid motion and pressure? (P2 - P1) /ρ + (u2

2- u12 )/2 = 0

Is the direction drawn here correct?

http://www.asknature.org/strategy/e27b89ebcdec8c9b5b2cd9ac84b8f8a0

Area 1 cm2 Blood velocity 10 cm/s

Aorta

Capillary Area 2 10-7 cm2

Blood velocity 0.1 cm/s

How many capillaries?

u1 = 0.2 m/s ; u2 = 3 m/s What is (P2 - P1) ?

Worksheet